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Using an ab initio density functional theory (DFT), we study thin film electronic properties of topological insulators (TIs) based on ternary compounds of Tl (thallium) and Bi (bismuth). We consider TlBiX2 (X=Se, Te) and Bi2X2Y (X, Y=Se, Te) compounds. Here we discuss the nature of surface states, their locations in the Brillouin Zone (BZ) and their interactions within the bulk region. Our calculations suggest a critical film thickness to maintain the Dirac cone which is smaller than that in binary Bi-based compounds. Atomic relaxations are found to affect the Dirac cone in some of these compounds. We discuss the penetration depth of surface states into the bulk region.

The nodal SDW order parameter on a cylindrical Fermi surface is thought to create the Dirac cone in the metallic ground state of iron pnictides and iron chalcogenides. Confirming appearance of the Dirac cone in DFT-GGA solutions of FeSe, we discuss origin of the bulk SDW order parameter in the stacked two-dimensional electronic system. In a layered system with vanishingly small inter-layer single-particle hopping processes, the exchange channels derived for the inter-plane magnetic interaction is the super-exchange counterpart of the two-particle Coulomb scattering for the pair-hopping channel in the layered superconductivity. The fluctuation reference method of the multi-reference density functional theory concludes existence of the inter-layer super-exchange interaction by the Coulomb off-diagonal elements among orbitals in the semi-metallic band structure. Thus a proof of 2D nature of the third generation Dirac cone in iron pnictides induced by SDW also promotes understanding of the high-temperature superconductivity.